Using spatial H₂ norm for sensor placement in parabolic partial differential equations

The issue of optimal sensor placement in the presence of disturbance is investigated for a class of transport-reaction processes, mathematically modeled by linear parabolic partial differential equations. Specifically, using modal decomposition to discretize the spatial coordinate, the optimal sensor location is computed through the solution of a nonlinear optimization problem in appropriate L₂ space. The notions of spatial and modal observability and robustness index are employed for the definition of the objective and constraint functionals. The formulated problem is subsequently solved using standard search algorithms. The proposed method is illustrated on a representative thermal diffusion process modeled by a one-dimensional parabolic PDE, where, in the presence of disturbances with known distribution, the optimal location of a single point sensor is computed.